Abstract
There still exist many unsolved problems on the study
related to John–Nirenberg spaces.
In this article, the authors introduce
two new vanishing subspaces of the John–Nirenberg space
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11971058
Award Identifier / Grant number: 12071431
Award Identifier / Grant number: 11871100
Funding source: Key Research and Development Program of China
Award Identifier / Grant number: 2020YFA0712900
Funding statement: This project is supported by the National Natural Science Foundation of China by Grant Nos. 11971058, 12071431 and 11871100, and also by the National Key Research and Development Program of China by Grant No. 2020YFA0712900.
Acknowledgements
Jin Tao would like to thank Yangyang Zhang and Hongchao Jia for some useful discussions on Lemma 4.2, Remark 5.9, and Proposition 5.7. The authors would also like to thank the referee for her/his carefully reading and several motivating remarks which indeed improve the quality of this article.
References
[1] D. Aalto, L. Berkovits, O. E. Kansanen and H. Yue, John–Nirenberg lemmas for a doubling measure, Studia Math. 204 (2011), no. 1, 21–37. 10.4064/sm204-1-2Search in Google Scholar
[2] A. C. Alabalik, A. Almeida and S. Samko, On the invariance of certain vanishing subspaces of Morrey spaces with respect to some classical operators, Banach J. Math. Anal. 14 (2020), no. 3, 987–1000. 10.1007/s43037-019-00049-7Search in Google Scholar
[3] A. C. Alabalik, A. Almeida and S. Samko, Preservation of certain vanishing properties of generalized Morrey spaces by some classical operators, Math. Methods Appl. Sci. 43 (2020), no. 16, 9375–9386. 10.1002/mma.6235Search in Google Scholar
[4] A. Almeida, Maximal commutators and commutators of potential operators in new vanishing Morrey spaces, Nonlinear Anal. 192 (2020), Paper No. 111684. 10.1016/j.na.2019.111684Search in Google Scholar
[5] A. Almeida and S. Samko, Approximation in Morrey spaces, J. Funct. Anal. 272 (2017), no. 6, 2392–2411. 10.1016/j.jfa.2016.11.015Search in Google Scholar
[6] A. Bényi, W. Damián, K. Moen and R. H. Torres, Compact bilinear commutators: The weighted case, Michigan Math. J. 64 (2015), no. 1, 39–51. 10.1307/mmj/1427203284Search in Google Scholar
[7] A. Bényi, J. M. Martell, K. Moen, E. Stachura and R. H. Torres, Boundedness results for commutators with BMO functions via weighted estimates: A comprehensive approach, Math. Ann. 376 (2020), no. 1–2, 61–102. 10.1007/s00208-019-01870-zSearch in Google Scholar
[8] A. Bényi and R. H. Torres, Compact bilinear operators and commutators, Proc. Amer. Math. Soc. 141 (2013), no. 10, 3609–3621. 10.1090/S0002-9939-2013-11689-8Search in Google Scholar
[9] L. Berkovits, J. Kinnunen and J. M. Martell, Oscillation estimates, self-improving results and good-λ inequalities, J. Funct. Anal. 270 (2016), no. 9, 3559–3590. 10.1016/j.jfa.2015.12.020Search in Google Scholar
[10] O. Blasco and C. Espinoza-Villalva, The norm of the characteristic function of a set in the John–Nirenberg space of exponent p, Math. Methods Appl. Sci. 43 (2020), no. 16, 9327–9336. 10.1002/mma.6124Search in Google Scholar
[11] A. Brudnyi and Y. Brudnyi, Multivariate bounded variation functions of Jordan–Wiener type, J. Approx. Theory 251 (2020), Article ID 105346. 10.1016/j.jat.2019.105346Search in Google Scholar
[12] A. Brudnyi and Y. Brudnyi, On the Banach structure of multivariate BV spaces, Dissertationes Math. 548 (2020), 1–52. 10.4064/dm801-7-2019Search in Google Scholar
[13] L. Chaffee, P. Chen, Y. Han, R. H. Torres and L. A. Ward, Characterization of compactness of commutators of bilinear singular integral operators, Proc. Amer. Math. Soc. 146 (2018), no. 9, 3943–3953. 10.1090/proc/14050Search in Google Scholar
[14]
D.-C. Chang, G. Dafni and E. M. Stein,
Hardy spaces, BMO, and boundary value problems for the Laplacian on a smooth domain in
[15] A. Clop and V. Cruz, Weighted estimates for Beltrami equations, Ann. Acad. Sci. Fenn. Math. 38 (2013), no. 1, 91–113. 10.5186/aasfm.2013.3818Search in Google Scholar
[16] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645. 10.1515/9781400827268.295Search in Google Scholar
[17]
G. Dafni,
Local VMO and weak convergence in
[18]
G. Dafni, T. Hytönen, R. Korte and H. Yue,
The space
[19] L. D’Onofrio, L. Greco, K.-M. Perfekt, C. Sbordone and R. Schiattarella, Atomic decompositions, two stars theorems, and distances for the Bourgain–Brezis–Mironescu space and other big spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire 37 (2020), no. 3, 653–661. 10.1016/j.anihpc.2020.01.004Search in Google Scholar
[20] L. C. Evans, Partial Differential Equations, Grad. Stud. Math. 19, American Mathematical Society, Providence, 1998. Search in Google Scholar
[21]
C. Fefferman and E. M. Stein,
[22] B. Franchi, C. Pérez and R. L. Wheeden, Self-improving properties of John–Nirenberg and Poincaré inequalities on spaces of homogeneous type, J. Funct. Anal. 153 (1998), no. 1, 108–146. 10.1006/jfan.1997.3175Search in Google Scholar
[23] R. Hurri-Syrjänen, N. Marola and A. V. Vähäkangas, Aspects of local-to-global results, Bull. Lond. Math. Soc. 46 (2014), no. 5, 1032–1042. 10.1112/blms/bdu061Search in Google Scholar
[24]
T. Iwaniec,
[25] S. Janson, Mean oscillation and commutators of singular integral operators, Ark. Mat. 16 (1978), no. 2, 263–270. 10.1007/BF02386000Search in Google Scholar
[26] H. Jia, J. Tao, D. Yang, W. Yuan and Y. Zhang, Special John–Nirenberg–Campanato spaces via congruent cubes, submitted. 10.1007/s11425-021-1866-4Search in Google Scholar
[27] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426. 10.1007/978-1-4612-5412-6_36Search in Google Scholar
[28] P. MacManus and C. Pérez, Generalized Poincaré inequalities: Sharp self-improving properties, Int. Math. Res. Not. IMRN 1998 (1998), no. 2, 101–116. 10.1155/S1073792898000099Search in Google Scholar
[29] N. Marola and O. Saari, Local to global results for spaces of BMO type, Math. Z. 282 (2016), no. 1–2, 473–484. 10.1007/s00209-015-1549-xSearch in Google Scholar
[30] M. Milman, Marcinkiewicz spaces, Garsia–Rodemich spaces and the scale of John–Nirenberg self improving inequalities, Ann. Acad. Sci. Fenn. Math. 41 (2016), no. 1, 491–501. 10.5186/aasfm.2016.4129Search in Google Scholar
[31] C. B. Morrey, Jr., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), no. 1, 126–166. 10.1090/S0002-9947-1938-1501936-8Search in Google Scholar
[32]
U. Neri,
Fractional integration on the space
[33] K.-M. Perfekt, Duality and distance formulas in spaces defined by means of oscillation, Ark. Mat. 51 (2013), no. 2, 345–361. 10.1007/s11512-012-0175-7Search in Google Scholar
[34] W. Rudin, Functional Analysis, 2nd ed., Int. Ser. Pure Appl. Math., McGraw-Hill, New York, 1991. Search in Google Scholar
[35] D. Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391–405. 10.1090/S0002-9947-1975-0377518-3Search in Google Scholar
[36] J. Sun, G. Xie and D. Yang, Localized John–Nirenberg–Campanato spaces, Anal. Math. Phys. 11 (2021), no. 1, Article ID 29. 10.1007/s13324-020-00445-5Search in Google Scholar
[37] J. Tao, D. Yang and D. Yang, Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces, Math. Methods Appl. Sci. 42 (2019), no. 5, 1631–1651. 10.1002/mma.5462Search in Google Scholar
[38] J. Tao, Q. Xue, D. Yang and W. Yuan, XMO and weighted compact bilinear commutators, preprint (2019), https://arxiv.org/abs/1909.03173. 10.1007/s00041-021-09854-xSearch in Google Scholar
[39] J. Tao, D. Yang and D. Yang, Beurling–Ahlfors commutators on weighted Morrey spaces and applications to Beltrami equations, Potential Anal. 53 (2020), no. 4, 1467–1491. 10.1007/s11118-019-09814-7Search in Google Scholar
[40] J. Tao, D. Yang and W. Yuan, John-Nirenberg-Campanato spaces, Nonlinear Anal. 189 (2019), Paper No. 111584. 10.1016/j.na.2019.111584Search in Google Scholar
[41] J. Tao, D. Yang and W. Yuan, A bridge connecting Lebesgue and Morrey spaces via Riesz norms, Banach J. Math. Anal. 15 (2021), no. 1, Paper No. 20. 10.1007/s43037-020-00106-6Search in Google Scholar
[42] J. Tao, D. Yang and W. Yuan, A survey on several spaces of John–Nirenberg type, submitted. Search in Google Scholar
[43] R. H. Torres and Q. Xue, On compactness of commutators of multiplication and bilinear pseudodifferential operators and a new subspace of BMO, Rev. Mat. Iberoam. 36 (2020), no. 3, 939–956. 10.4171/rmi/1156Search in Google Scholar
[44] A. Uchiyama, On the compactness of operators of Hankel type, Tohoku Math. J. (2) 30 (1978), no. 1, 163–171. 10.2748/tmj/1178230105Search in Google Scholar
[45] C. T. Zorko, Morrey space, Proc. Amer. Math. Soc. 98 (1986), no. 4, 586–592. 10.1090/S0002-9939-1986-0861756-XSearch in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Borderline regularity for fully nonlinear equations in Dini domains
- Optimal Łojasiewicz–Simon inequalities and Morse–Bott Yang–Mills energy functions
- Generalized principal eigenvalues of convex nonlinear elliptic operators in ℝ N
- A note on Kazdan–Warner equation on networks
- Integral representation for energies in linear elasticity with surface discontinuities
- Minkowski inequalities and constrained inverse curvature flows in warped spaces
- Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian
- Pairings between bounded divergence-measure vector fields and BV functions
- Equivalence between distributional and viscosity solutions for the double-phase equation
- Vanishing John–Nirenberg spaces
- Extremals in nonlinear potential theory
- On the structure of divergence-free measures on ℝ2
- Solutions of the (free boundary) Reifenberg Plateau problem
- Equilibrium measure for a nonlocal dislocation energy with physical confinement
Articles in the same Issue
- Frontmatter
- Borderline regularity for fully nonlinear equations in Dini domains
- Optimal Łojasiewicz–Simon inequalities and Morse–Bott Yang–Mills energy functions
- Generalized principal eigenvalues of convex nonlinear elliptic operators in ℝ N
- A note on Kazdan–Warner equation on networks
- Integral representation for energies in linear elasticity with surface discontinuities
- Minkowski inequalities and constrained inverse curvature flows in warped spaces
- Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian
- Pairings between bounded divergence-measure vector fields and BV functions
- Equivalence between distributional and viscosity solutions for the double-phase equation
- Vanishing John–Nirenberg spaces
- Extremals in nonlinear potential theory
- On the structure of divergence-free measures on ℝ2
- Solutions of the (free boundary) Reifenberg Plateau problem
- Equilibrium measure for a nonlocal dislocation energy with physical confinement